What are some practical uses of matrices?

Well...I'm not sure what your concept is of the "everyday world"...but it can be said that electrical engineers make constant use of what are known as "transfer matrices" in "black box analysis" ....that is, you have a certain voltage and current signal going into the "black box" (amplifier, synthesizer, filter, whatever) and at the output terminals you have an output voltage and current...the transfer matrix will tell you what output voltage and current you will get for a given input voltage/current pair...and the transfer matrix is usually a simple 2 x 2 square matrix.

Also....matrices (rectangular this time) could be used in marketing or pricing analysis...supoose you ran a shoe store and you offered three different styles of shoes...and you priced each style differently the first week, then changed the prices the second week...again the third, etc. for four weeks. Well, then you could enter those figures in a 3 x4 ( 3 columns and four rows) rectangular matrix. OK now suppose you kept track of exactly how many pairs of each style you sold in each of the four weeks...then you could tabulate each of those figures in a 4 x 3 (four columns, three rows this time ) rectangular matrix as well. Now suppose you multiplied those two matrices together by the usual method...you would then have a 4 x 4 square matrix each element of which would give you the total amount you either took in in any given week or WOULD have taken in in one week, charging the prices in effect during another week----specifically all the numbers in the diagonal of the 4 x 4 would tell you what you DID make in the first, second, etc week and all the OFF DIAGONAL numbers would tel you what you WOULD have taken in in one week charging the prices of another week. So matrices can be used as a marketing analysis tool.

Always remember never to confuse a matrix with a tensor...many authors, especially in the exact sciences use the terms interchangeably and they know the difference and they are assuming that you do too...a matrix and a tensor are two different things but a matrix CAN very conveniently be used to symbolize or display the components of either a first or second rank tensor. Always remember to keep the two distinct in your mind because matrices can be used where tensors would have no meaning and visa versa----especially in theoretical physics as regards tensors which can have any rank.

This is a very broad topic, matrices are very abstract in the terms of how we can implement them in different strands of everday life. One example I can give you is in Computer Graphics, matrices can be used to represent transformations of objects, images or points.